gamma-pareto distribution and its applications

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Journal of Modern Applied Statistical Methods

Volume 11 | Issue 1 Article 7

5-1-2012

Gamma-Pareto Distribution and Its Applications Ayman Alzaatreh Austin Peay State University, Clarksville, TN

Felix FamoyeCentral Michigan University

Carl LeeCentral Michigan University

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Recommended Citation Alzaatreh, Ayman; Famoye, Felix; and Lee, Carl (2012) "Gamma-Pareto Distribution and Its Applications," Journal of Modern Applied Statistical Methods: Vol. 11: Iss. 1, Article 7.

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ALZAATREH, FAMOYE & LEE

79

( )( ) ( )1

1 1

( )

( ) log 1 ( ) 1 ( ).

( )

g x

f x F x F xα

β

α α β

− −

=

− − −

Γ

(1.3)

When 1 β = , the gamma- X family in (1.3)

reduces to

( )( )11

( ) ( ) log 1 ( ) .( )

g x f x F x α

α

−

= − −Γ

(1.4)

Alzaatreh, et al. (2012) noted that, when nα =

where n is a positive integer, the distribution in(1.4) can be written as

( )( )11

( ) ( ) log 1 ( )( 1)!

n

g x f x F xn

−

= − −−

,

which is the density function of the upper record

values,( )U n X , arising from a sequence

1

n

i i X

=

of identically independent random variables with

PDF ( ) f x and CDF ( ) F x (Johnson, et al.,

1994).

The Gamma-Pareto Distribution If X is a Pareto random variable with

density function1( ) / ,k k f x k x xθ θ +

= > ,

then (1.3) results in

1/

( ) log ,( )

.

k k x

g x x x

x

α β α

α

θ

α β θ

θ

−

=

Γ

>

(2.1)

Setting / k c β =

, (2.1) reduces to

11/1

( ) log ,( )

, , 0; .

c x

g x x c x

c x

α

α

θ

α θ

α θ θ

−

=

Γ

> >

(2.2)

A random variable X with the PDF g ( x)in (2.2) is said to follow the gamma-Pareto

distribution. When 1α = the gamma-Pareto

distribution reduces to the Pareto distribution

with parameters 1/c and θ , and when 1nα = + ,

it reduces to the upper record value distributionarising from Pareto identically independentrandom variables (Ahsanullah & Houchens,1989). From (2.2), the CDF of the gamma-Pareto distribution is obtained as

( ) 1( ) , log / / ( )G x c xγ α θ α −= Γ ,

(2.3)

where 1

0( , )

t ut u e duα γ α − −

= is the incomplete

gamma function.

Johnson, et al. (1994) discusseddifferent types of Pareto distributions and theirCDFs. These are

Using equation (1.3), the corresponding PDF ofgamma-Pareto II, gamma-Pareto III, andgamma-Pareto IV can be written as shown inTable 1.

Some relationships among thesedistributions are:

• If Y follows the gamma-Pareto distribution

in (2.2), then the translation X Y θ = −

follows the gamma-Pareto II distribution.

• When a = 1 and b = 0, the gamma-Pareto IIIdistribution reduces to the gamma-Pareto IIdistribution.

ParetoII:

( ) 1 , 0.( )

a

a F x x

x

θ

θ = − >

+

ParetoIII:

( ) 1 , 0.( )

bx

a

e F x x

x

θ

θ

−

= − >+

ParetoIV:

1/

( ) 1 1 , .a

x F x x

γ µ µ

σ

− − = − + >



GAMMA-PARETO DISTRIBUTION AND ITS APPLICATIONS

80

• When 1γ = and µ σ θ = = , the gamma-

Pareto IV distribution reduces to the

gamma-Pareto distribution in (2.2) with parameters α , c and θ .

• When 1γ = and 0 µ = , the gamma-Pareto

IV distribution reduces to the gamma-ParetoII distribution.

Properties of the gamma-Pareto distribution

The following Lemma shows therelationship between the gamma-Paretodistribution and the gamma distribution.

Lemma 1If a random variable Y follows the

gamma distribution with parameters α and c,

then the random variableY X eθ = follows the

gamma-Pareto distribution with parameters α ,c and θ .

Lemma 1 ProofThe result follows by using the

transformation technique.The hazard function associated with the

gamma-Pareto distribution is

( )

1/ 1

1 1/ 1

( )( )

1 ( )

(log( / )),

( ( ) , log / )

,

c

c

g xh x

G x

x

x c c x

x

α

α

θ θ

α γ α θ

θ

−

+ −

=−

=Γ −

>

(3.1)

and the limiting behaviors of the gamma-ParetoPDF and the hazard function are given in thefollowing theorem.

Theorem 1The limit of the gamma-Pareto density

function and the gamma-Pareto hazard function

Table 1: Corresponding PDFs of gamma-Pareto Distributions

gamma-Pareto II:

1/1 1/ 1( ) ( ) [log(1 / )] , 0, where .

( )

cc g x x x x c

c a

α

α

θ β θ θ

α

− − −= + + > =

Γ

(2.4)

gamma-Pareto III:

11/

1( ) log , 0.

( ) ( ) ( )

bx bx

a a

a e e g x b x

x x x

α β

α

θ θ

β α θ θ θ

−− −

= + − > Γ + + +

(2.5)

gamma-Pareto IV:

11 1/

1 1/ 1/ 1/1( ) 1 log 1 ,( )

, where / .

c

x x x g x

c

x c a

α

γ γ γ

α

µ µ µ γσ α σ σ σ

µ β

−− −

− + − − − = + + Γ

> =

(2.6)




81

as x → ∞ is 0 and the limit as x θ +→ is given

by

0, 1

lim ( ) lim ( ) 1/ ( ), 1

, 1. x x

g x h x c

θ θ

α

θ α

α + +

→ →

>

= = =

∞ <

(3.2)

Theorem 1 Proof

First it can be shown that lim ( ) x

g x→∞

= 0. If

1α ≤ , then from definition (2.2) lim ( ) 0 x

g x→∞

= ,

and if 1α > , then

11/

1

1/( 1)

lim ( )

( / ) 1 lim lim log

( )

0 lim log / .

x

c

x x

x

g x

x x

c x

x x

α

α

α

α

θ

α θ

θ

→∞

−

→∞ →∞

−

−

→∞

= ×

Γ

= ×

Using L’Hôpital’s rule, it can be shown that

( )1

1/( 1)lim log( / ) / 0 x

x xα

α θ −

−

→∞= .

To show that lim ( ) 0 x

h x→∞

= , we have

lim ( ) lim ( ) / (1 ( )) x x

h x g x G x→∞ →∞

= − . Because

lim ( ) 0 x

g x→∞

= , L’Hôpital’s rule can be applied

and implies that

lim ( ) lim ( ) / ( )

1 1lim

log( / )

0.

x x

x

h x g x g x

c

x x cx

α

θ

→∞ →∞

→∞

′=

− += −

=

The result in (3.2) follows directly from the

definition of (2.2) and ( ) ( ) / (1 ( ))h x g x G x= − .

The following theorem shows that the gamma-Pareto distribution is unimodal.

Theorem 2The gamma-Pareto distribution has a

unique mode at 0 x x= . When 1α ≤ the mode

is 0 x θ = and when 1α > the mode is

( 1)/( 1)

0 ec c x α θ − += .

Theorem 2 ProofThe derivative with respect to x of

equation (2.2) is given by

1/ 2 1/

2

( ) ( )

log ( 1) ( 1)log .

c c g x x

x xc c

α

θ α

α θ θ

−

−

′ = Γ

× − − +

(3.3)

From (3.3) the critical points of ( ) g x are x = θ

and( 1)/( 1)ec c x

α θ − += . For 1α ≤ , it may be

observed from (3.3) that ( ) 0 g x′ < , therefore

( ) g x is strictly decreasing. Also, from Theorem

1, lim ( ) 1/ ( ) x

g x cθ

θ +

→

= when 1α = and

lim ( ) x

g xθ +→

= ∞ when 1α < . Thus, ( ) g x has a

unique mode at x = θ . Using Theorem 1, for

1α > , lim ( ) 0 x

g xθ +→

= implies that x θ =

cannot be a modal point, hence the mode is( 1)/( 1)

0 ec c x α θ − += .

Graphs of ( ) g x and ( )h x are

displayed in Figures 1-3. The plots show that thegamma-Pareto PDF has a very long right tail and

also that when parameters c and θ increase, the

peak of the distribution decreases. In addition,the graphs of the gamma-Pareto PDF indicate

that ( ) g x is a right skewed distribution. The

plots in Figure 3 illustrate that the gamma-Paretohazard function is either monotone decreasing orupside-down bathtub.

The entropy of a random variable X is ameasure of variation of uncertainty (Rényi,1961). Shannon entropy (Shannon, 1948) for arandom variable X with PDF g ( x) is defined as

( ) log ( ) E g X − . Shannon showed important

applications of this entropy in communicationtheory and many applications have been used in




82

Figure 1: The gamma-Pareto PDF for Various Values of α, c and θ

Figure 2: The gamma-Pareto PDF for Various Values of α, c and θ

1 2 3 4 5 6

0 . 0

0 . 5

1 . 0

1 . 5

2 . 0

x

g ( x )

alpha =0.5, c =1, theta =1

alpha =0.5, c =0.1, theta =1

alpha =1, c =1, theta =1


2 4 6 8 10

0 . 0

0 . 1

0 . 2

0 . 3

0

. 4

0 . 5

x

g ( x )


alpha =2, c =0.5, theta =1

alpha =10,c =0.15, theta =1






83

different areas such as engineering, physics, biology and economics.

Lemma 2The Shannon entropy of the gamma- X

family of distributions is given by

( )( ) 1log 1 (1 )

log log ( ) (1 ) ( ),

T

X E f F eη α β

β α α ψ α

− −= − − + −

+ + Γ + −

where ψ is the digamma function and T is the

gamma random variable with parameters α and

β .

Lemma 2 ProofSee Alzaatreh, et al. (2012) for proof

details.

Theorem 3The Shannon entropy for the gamma-

Pareto distribution is given by

log log log ( )

( 1) (1 ) ( ).

X c

c

η θ α

α α ψ α

= + + Γ

+ + + −

(3.4)

Theorem 3 ProofFirst it is necessary to find

( )( ) 1log 1 T E f F e− −− − , where

1( ) / .k k f x k xθ += It can be shown that

1 1/( ) (1 ) k F x xθ − −= − , thus

( )( ) 1log 1

log log (1 1/ ) ( )

T E f F e

k k E T θ

− −− − =

− + +

The result follows from Lemma 2 by noting that

( ) E T αβ = and /c k β = (see equation 2.1).

The Rényi (1961) entropy for the

random variable X with PDF ( ) g x is defined as

Figure 3: The gamma-Pareto Hazard Function for Various Values of α, c and θ

0 5 10 15 20 25 30

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

x

h ( x )




alpha =10, c =0.15,theta =2




84

1

( ) log ( ) , 0, 1.1

s

R I s g x dx s s

s= > ≠

−

(3.5)

By using the gamma-Pareto PDF in (2.2), we

have

( 1)/

( ) ( )

log .

s s s

s s c

s

g x dx c

x x dx

x

α

θ

α

θ

α

θ

θ

∞− −

−∞

−

= Γ

×

(3.6)

Substituting log( / )u x θ = , (3.6) can be re-

written as

1

( 1) 1

( ( 1) 1)( )

( )[ (1 1/ ) 1]

s s

s s s

s g x dx

c s cα α θ

θ α

α

−∞

− +

Γ − +=

Γ + − .

(3.7)

Using equation (3.7), the Rényi entropy in (3.5)can be written as

11 1

log log ( )( ) log

log ( ) log( (1 ) R s

s c s I s

s c

α α θ

ξ ξ − −

+ Γ = −

− Γ + +

(3.8)

where ( 1) 1 sξ α = − + . Shannon entropy is a

special case of Rényi entropy obtained by taking

the limit of Rényi entropy as 1 s → . The result

in (3.4) follows by using the L’Hôpital’s rule for

evaluating the limit of equation (3.8) as 1 s → .

Moments and Mean Deviations

The non-central moments for thegamma-Pareto distribution in (2.2) can be

written as

1

11/

1

( ) ( )

log .

r

c

r

E X c

xdx

x

α

α

θ

α

θ

θ

− −

−∞

−

= Γ

×

(4.1)

Substituting log( / )u x θ = reduces (4.1) to

( ) (1 ) , 1/r r E X rc c r α θ −

= − < . (4.2)

Hence, the mean for the gamma-Pareto

distribution is

(1 ) , 1c cα µ θ −

= − < .

Note that when 1α = in equation (4.2),1( ) (1 )r r E X rcθ −

= − which represents the non-

central moments for the Pareto distribution with

parameters 1/c and θ .Using the binomial expansion for

( )r X µ − , the central moments ( )r E X µ − for

any random variable X can be written as

0

( ) ( 1) ( )r

r r k r k k

k

r E X E X

k µ µ − −

=

− = −

.

(4.3)

Using equations (4.2) and (4.3), the centralmoments for the gamma-Pareto random variable X can be simplified to

( )

0

( )

( 1) (1 ) (1 ) .

r

r

r r k k r

k

E X

r c kck

α α

µ

θ − − −

=

− =

− − −

(4.4)

Note that equation (4.4) indicates that the centralmoments of the gamma-Pareto distribution is an

increasing function of θ . Using (4.4), the

variance, the skewness and the kurtosis for thegamma-Pareto distribution are respectivelyexpressed as

2 2 2

[(1 2 ) (1 ) ], 0.5c c cα α

σ θ − −

= − − − < (4.5)

3

2 3/1 2

(1 3 ) 2(1 ) 3(1 2 ) (1 )

((1 2 ) (1 ) )

c c c c

c c

α α α α

α α γ

− − − −

− −

− + − − − −

− − −=

(4.6)




86

The deviation from the mean and themedian are used to measure the dispersion andspread in a population from the center. If themedian is denoted by M , then the mean

deviation from the mean , ( ) D µ , and the mean

deviation from the median, ( ) D M , can be

written as

( ) ( ) ( ) ( ) ( )

2 ( ) ( )

2 ( ) 2 ( ) .

D x g x dx x g x dx

x g x dx

G xg x dx

µ

θ µ

µ

θ

µ

θ

µ µ µ

µ

µ µ

∞

= − + −

= −

= −

(4.6)

( ) ( ) ( ) ( ) ( )

2 ( ) ( ) ( )

2 ( ) 2 ( )

2 ( ) .

M

M

M

M

M

D M M x g x dx x M g x dx

M x g x dx E X M

MG M M xg x dx

xg x dx

θ

θ

θ

θ

µ

µ

∞

= − + −

= − + −

= + − −

= −

(4.7)

Consider the integral

1/ 1

( )

1 ( / ) (log( / )) .

( )

m

mc

xg x dx

x x dxc

θ

α

α θ θ θ

α −

=

Γ

(4.8)

Using the substitution log( / )u x θ = , the

equation (4.8) can be written as

1( ) [ , ( 1) log( / )]( )

m

xg x dx c mθ

µ γ α θ

α −

= −Γ ,

(4.9)

and by using equations (2.3) and (4.9), the meandeviation from the mean is

1

1

[ , log( / )]2

[ , ( 1) log( / )]( ) , 1,

( )

c

c D c

γ α µ θ µ

γ α µ θ µ

α

−

−

− − = <

Γ

and the mean deviation from the median is

2 1( ) 1 , 1 log , 1.

( )

M D M c

c µ γ α

α θ

= − − <

Γ

Parameter Estimation

When 1α < , the likelihood function for

the gamma-Pareto distribution goes to infinity as

θ approaches the sample minimum(1)

; thus,

when 1α < and θ is estimated by(1) x , no MLE

for α and c exists. A similar problem was

studied by Smith (1985) who proposed analternative approach for estimating the parameters as follows: If sample data

1 2, ,..., n x x are observed, estimate the

parameter θ by the sample minimum(1) x and

then use the MLE method to estimate α and c

by excluding the sample minimum.Applying Smith’s method to obtain the

MLE for the gamma-Pareto parameters, the log-likelihood function for the gamma-Pareto

distribution is given by

( )

(1)

(1)

(1)

1

(1)

(1) (1)

log ( , ) log ( ; , , )

log log ( ) log (1 )

log( / ) ( 1) log log( / ) .

i

i

i

x x

x x

i i

L c g x x c

c x c

x x x x

α α

α α

α

≠

−

≠

=

= − − Γ − − +

× + −

(5.1)

The derivatives of (5.1) with respect to α and c

are given by

(1) (1)

loglog ( ) log log

i

i

x x

x Lc

xψ α

α ≠

∂ = − − − ∂

(5.2)




87

(1)

2

(1)

log 1log

i

i

x x

x L

c c c x

α

≠

∂ − = + ∂

.

(5.3)

Setting (5.3) to zero and simplifying results in

(1)

(1)

1log( / )

( )i

i

x x

c x xn n α ≠

=′− ,

(5.4)

where n′ is the frequency of(1)

. Equation

(5.4) can be written as

*

1

1c m

α = , (5.5)

where

(1)

*

1 (1)log( / ) / ( )i

i

x x

m x x n n≠

′= − is the

sample mean for(1)log( / )i x after excluding

(1).

Setting (5.2) to zero and using equation(5.5) results in

* *

1 2( ) log( ) log( ) 0m mψ α α − + − = ,

(5.6)

where

(1)

*

2 (1)log(log( / )) / ( )i

i

x x

m x x n n≠

′= − is

the sample mean for(1)log(log( / ))i x x after

excluding(1) x . The MLE α of α is the

solution of equation (5.6) and the MLE c of c

can be determined by substituting the estimate

α in equation (5.5).

The initial values for the parameters α and c can be obtained by assuming the random

sample (1)log( / ), 1,...,i iY X x i n= =

are takenfrom the gamma distribution with parameters α and c. By equating the population mean and the population variance of gamma distribution (with parameters α and c) to the corresponding

sample mean and sample variance of

, 1,...,i i n= and then solving for α and c,

the initial values are2

0 / yc s y= and

2 2

0 / y y sα = , where 2

y s and y are the sample

variance and the sample mean for 1 , 2 , …,

n .

Lemma 3The Fisher information matrix for the

gamma-Pareto distribution when θ is known isgiven by

2

( ) 1/

1/ /

c I n

c c

ψ α

α

′ =

. (5.7)

Lemma 3 ProofThe Fisher information matrix is defined

by [ ]ij I I = with

( )2

log ( , )ij i

i j

I E L x τ τ τ

−∂ =

∂ ∂

, where 1τ α =

and2 cτ = . To find I , the second derivatives of

1log ( , ) log ( ; , , )

n

ii L c g x cα θ α

== , are

needed. These can be obtained from the

derivatives of (5.2) and (5.3), where(1) x is

replaced by θ and the sums are taken form

1i = to n. The second derivatives of

log ( , ) L cα can be written as2 2log / ( ) L nα ψ α ′∂ ∂ = − ,

2 log / / L c n cα ∂ ∂ ∂ = − ,

and2 2 2 3

1log / 2 log( / ).

n

ii L c n c c xα θ − −

=∂ ∂ = −

From Lemma 1, it may be concluded that

(log( / )) E X cθ α = . The results of (5.7) follow

from taking the negative expected values of thesecond derivatives.

Theorem 5The variance-covariance matrix for the

gamma-Pareto distribution when θ is known is

given by




88

2

1

( ( ) 1) ( )

c

n c c

α

αψ α ψ α

− Σ =

′ ′− − .

(5.8)

Theorem 5 Proof

The result follows by taking the inverseof the Fisher information matrix in (5.7). From

(5.8), the variance of the ML estimates α and

c are respectively given by

2

ˆvar( )( ( ) 1)

and

( )ˆvar( ) .

( ( ) 1)

n

cc

n

α α

αψ α

ψ α

αψ α

=′ −

′=

′ −

(5.9)

From (5.9), the variance of α does not depend

on the parameter c. Also, as c increases the

variance of c increases. Using the

approximation1 2 31 1

( )2 6

ψ α α α α − − −′ + +

(Johnson, et al., 1994, page 357), equations (5.9)can be approximated as

3

2 2

6ˆvar( )

(3 1)and

(6 3 1)ˆvar( ) .

(3 1)

n

cc

n

α α

α

α α

α α

+

+ +

+

(5.10)

Based on the Central Limit Theorem,

ˆ ˆ( ) / ( ) (0, 1)d se N α α α − → and

ˆ ˆ( ) / ( ) (0, 1)d c c se c N − → where ˆ( ) se α and

ˆ( ) se c are the standard errors of α and c

respectively.In the following, the uniformly

minimum variance unbiased estimator(UMVUE) is derived for the parameter c

assuming that the parameters α and θ are

known. The following theorem by Lehmann &Scheffé (1950) is needed in order to find theUMVUE for the parameter c.

Theorem 6

Let1 2, ,...,

n X X X be a random sample

from PDF ( , ), . g x β β ∈ Ω Let T be a sufficient

statistic for β and let the family

( , ), g T β β ∈ Ω of probability densityfunctions be complete. If there is a function of T

that is an unbiased estimator of β , then this

function of T is the UMVUE of β .

Lemma 4

If α and θ are known for the gamma-

Pareto distribution, then1log( / )

n

iiT x θ

== is

a sufficient statistic for the parameter c.

Lemma 4 ProofLet

1 2, ,..., n X X X be a random sample

from the gamma-Pareto distribution, the jointdensity function is then given by

1/ 1

1 2

1

1 2

( / ) (log( / ))( , ,..., | )

( )

( , ) ( , ,..., )

cni i

n

i i

n

x x g x x x c

x c

k c T h x x x

α

α

θ θ

α

−

=

=Γ

=

∏

where 1( , ) exp log (1 )k c T n c c T α −= − + + ,

1 2( , ,..., ) exp log( ( ))

( 1) log(log( / ))

n

i

h x x x

x

θ α

α θ

= − Γ

+ −

and1log( / )

n

iiT x θ

== . Thus, by using the

factorization theorem, the statistic T is sufficientfor the parameter c because the parameters α and θ are known.

Theorem 7

If α and θ are known for the gamma-

Pareto distribution, then1

1 1( ) log( / )

n

iim n xα θ −

== is the UMVUestimator for the parameter c.

Theorem 7 ProofBy using Lemma 4,

1log( / )

n

iiT x θ

== is a sufficient statistic for

c. It follows from Lemma 1 that




89

1log( / )

n

iiT x θ

== has a gamma distribution

with parameters nα and c . Because gamma

density belongs to the exponential family, this

implies that ( , ), 0 g T c c > is a complete

family where ( ) g x is the gamma-Pareto

density. Also from Lemma 1, 1( ) E m c= , hence

1m is an unbiased estimator for parameter c. By

applying Theorem 6, the statistic 1m is the

UMVUE of c. From equation (5.5), it is

interesting to note that when θ ( (1)= ) and α

are known, the MLE of c is the UMVUE of parameter c.

ApplicationsThe gamma-Pareto is applied to three

data sets: The first data set (see Table 3) wasanalyzed by Akinsete, et al. (2008) andrepresents Floyd River flood rates for the years1935-1973 in Iowa, USA. The second data set(see Table 5) is from Mahmoudi (2011) and itrepresents the fatigue life of 6061-T6 aluminumcoupons cut parallel with the direction of rollingand oscillated at 18 cycles per second. The thirddata set (see Table 7) was analyzed by Eugene(2001) and represents the observed frequenciesfor Tribolium Confusum Strain #3. Themaximum likelihood estimates, the log-

likelihood value and the AIC (AkaikeInformation Criterion) values for the fitteddistributions are reported in Tables 4, 6 and 8.

Akinsete (2008) fitted the data in Table3 to the beta-Pareto distribution and comparedthe result with the Pareto and the generalized

Pareto distribution (Pickands, 1975). Results areshown in Table 4, along with the result obtained by fitting the gamma-Pareto distribution to thedata. The results show that both beta-Pareto andgamma-Pareto distributions provide adequate fitto the data. Because the gamma-Pareto

distribution has only three parameters, this is anadvantage for using it over the four-parameter beta-Pareto distribution. In examining thedistribution of this data, observe that the data hasa reversed J-shape distribution; this suggests thatthe gamma-Pareto distribution performs well inmodeling reversed J-shape distribution. Figure 4displays the empirical and the fitted cumulativedistribution functions and supports the resultsshown in Table 4.

Mahmoudi (2011) proposed a five- parameter beta generalized Pareto distribution.

He fitted the data (shown in Table 5) andcompared the result with beta-Pareto, three- parameter generalized Pareto, Weibull andPareto distributions. To conserve space, only theresults of fitting beta generalized Pareto and beta-Pareto from Mahmoudi (2011) are reportedin Table 6 along with the result of fitting thegamma-Pareto distribution to the data. Theresults in Table 6 indicate that the gamma-Paretodistribution provides the best fit among thedistributions. The distribution of this dataindicates that the data is approximately

symmetric. This example suggests that thegamma-Pareto distribution does very well infitting the distribution of data which isapproximately symmetric. Figure 5 displays theempirical and the fitted cumulative distributionfunctions and supports the results shown inTable 6.

Table 3: Annual Flood Discharge Rates of the Floyd River Data

Years Flood Discharge (ft3 / s)

1935-1944 1460 4050 3570 2060 1300 1390 1720 6280 1360 7440

1945-1954 5320 1400 3240 2710 4520 4840 8320 13900 71500 6250

1955-1964 2260 318 1330 970 1920 15100 2870 20600 3810 726

1965-1973 7500 7170 2000 829 17300 4740 13400 2940 5660




90

Table 4: Parameter Estimates for the Floyd River Flood Data

DistributionParameterEstimates

LogLikelihood

AIC

Paretok = 0.4125

q = 318-392.81 789.62

GeneralizedPareto

k = -0.3071

q = 4520-379.55 763.09

Beta-Pareto

a = 6.1550

b = 24.2434

k = 0.0926

ˆq = 318

-365.45 738.9

Gamma-Pareto

a = 5.1454

c =0.4712

q = 318

-365.81 734.9

Figure 4: CDF for Fitted Distributions for Floyd River Flood Data

0 10000 20000 30000 40000 50000 60000 70000

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

Empirical

Beta-Pareto

ParetoGeneralized Pareto

Gamma-Pareto

Flood Discharge

C D F




92

Eugene (2001) proposed the beta-normaldistribution and termed it the generalized normaldistribution. Eugene (2001) fitted the data inTable 7 and compared the result with gammadistribution and Lagrange-gamma distribution proposed by Famoye and Govindarajulu (1998).

These results are reported in Table 8 along withthe result of fitting the data to the gamma-Paretodistribution. The results from the log-likelihoodand AIC values indicate that the gamma-Paretoand the generalized normal distributions fit thedata best. Figure 6 displays the empirical and thefitted cumulative distribution functions. Figure 6shows that the generalized normal distributiondoes not fit the left tail very well, however, thegamma-Pareto distribution does provide a goodfit. The distribution shows that the data has along right tail. This example suggests that the

gamma-Pareto distribution does very well infitting the distributions of data with a long righttail characteristic.

ConclusionThis article defined the gamma- X family andstudied a special case of the gamma- X family,the gamma-Pareto distribution. Various properties of the gamma-Pareto distributionwere investigated, including moments,

deviations from the mean and median, hazardfunction, unimodality, entropies and Fisherinformation matrix. Results of the uniformlyminimum variance unbiased estimator wasobtained for one of the shape parameters of thegamma-Pareto distribution. Three real data setswere fitted to the gamma-Pareto distribution andcompared with other known distributions.Results show that the gamma-Pareto distribution provides a good fit to each data set and suggeststhat the gamma-Pareto distribution can be agood model to fit data with a reversed J-shape,

approximately symmetric and long right tailcharacteristics.

Figure 5: CDF for Fitted Distributions for Fatigue Life of 6061-T6 Aluminum Data

80 100 120 140 160 180 200

0 . 0

0 . 2

0 . 4

0 . 6

0 .

8

1 . 0

Empirical

Beta-Pareto

Pareto

Beta-Generalized Pareto

Gamma-Pareto

fatigue life of 6061-T6 aluminum

C D F




93

Table 7: Observed frequencies for Tribolium Confusum Strain # 3

x-Values Frequency x-Values Frequency x-Values Frequency

55 3 125 51 195 1

65 20 135 20 205 2

75 53 145 11 215 0

85 78 155 6 225 1

95 86 165 4 235 1

105 86 175 7 245 1

115 68 185 5

Table 8: Parameter Estimates for the Tribolium Confusum Strain # 3 Data

DistributionParameterEstimates

LogLikelihood

AIC

Gammaa = 15.15

b = 6.92-2335.31 4674.62

Lagrange-gamma

r = 31ˆl=0.36842

q = 0.02913

-2314.2 4640.41

Generalized Normal

a = 28.68

b = 0.20

m = 30.65

s = 22.04

-2290.85 4597.71

Gamma-Pareto

a = 6.3513c =0.09743

q = 55

-2297.7 4599.4




94

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Akinsete, A., Famoye, F., & Lee, C.(2008). The beta-Pareto distribution, Statistics,

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Figure 6: CDF for Fitted Distributions for Tribolium Confusum Strain # 3 Data

50 100 150 200 250

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

Empirical

Gneralized Normal

Gamma

Gamma-Pareto

Lagrange Gamma

Tribolium Confusum Strain

C D F

gamma-pareto distribution and its applications

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